I've been wanting to do some Let's Plays for a while and actually recorded some footage recently, but the biggest problem is I don't have something to edit with. Okay yes Windows Movie Maker exists, but let's be honest, does it really count?

So what I need is any editor I can get for free that can do the following:

Honestly, if all you really want to do are the three things listed then Windows Movie Maker is probably more than sufficient (at least when combined with an audio editor to clear up and improve the audio; try Wavosaur or Audacity).

If you really do want more powerful editing then something like Lightworks or DaVinci Resolve both have free versions available. DaVinci Resolve is primarily a colour correction suite but its got a powerful enough set of tools and you don't lose a huge amount compared to the paid version. The free version of Lightworks is a bit more limited as while you get almost all the tools the free version limits your output formats and restricts you to 720p. In both cases you're getting a powerful video editor with a lot of features but neither are particularly user friendly and if you only want to do the basic things you suggest there's going to be a lot of things within them that you don't use.

While it's not free you may want to consider looking at Adobe Premiere. Their move to a subscription based service means that instead of a one-off huge cost you're instead paying around $20 a month. For that you get a very powerful video editor with a huge amount of features, one that's very user friendly and, because of how popular it is, there's a vast amount of support and tutorials out there. You likely don't need it for what you're intending to do but After Effects may be worth a look as well; Premiere deals more with editing and composing long videos while After Effects is about the detailed editing of shorter clips and creating motion graphics (such as logos).